Modelling advection with MOHID

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Modelling advection with MOHID
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Contents

• What is advection?
• The importance of advection in coastal areas
• Methodology implemented in MOHID to simulate
  advection
• Code trip
• Test cases
• 1D and 2 D cyclic channel
• Symetric estuary
• Freshwater cylinder
• Density front

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What is advection ?
k
i
j
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The importance of advection in coastal waters

• Estuary mouths horizontal advection of
  momentum
• Coastal Upwelling vertical and horizontal
  advection of heat
• Baroclinic instabilities horizontal advection
  of mass, of heat and of momentum

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Aveiro Mouth
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Upwelling in schematic coast
Schematic slope
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(No Transcript)
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Upwelling scenario more important because is
the scenario more common in summer
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Density front
Ajustamento geostrófico segundo um modelo
analítico baseado nas equações de gravidade
reduzida a) Instante inicial (velocidade
perpendicular à frente) b) instante em que é
atingido o equilíbrio velocidade paralela à
frente.
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Baroclinic instability

• Vorticidade potencial (explicação de oceanógrafo)

w
(fw)/H const.
H
H
w

• Balanço de forças (explicação de engenheiro)

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Modelling a density front with Mohid

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Modelling a density front with Mohid
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Some details
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Methodology implemented in Mohid to compute
advection
Central Differences
2nd order upwind (1D Quick)
3th order upwind (1D Quickest)
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Problems related with higher order schemes
Ok
Vi
Vi-1gtgtVi Or Vi-1ltltVi
Problems
?0
Vi-1
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Problems related with higher order schemes

• Lack of positivity production of ripples near
  sharp gradients

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Total Variation Diminishing

• TVD schemes implemented in Mohid
• if rgt0
• MinMod ?min(1,r)
• Van Leer ?2r/(1r)
• MUSCL min(2,2r,(1r)/2)
• SuperBee max(min(1,2r),min(r, 2))
• PPM max(min(Aux,2/(1-Cr),2r/Cr))
• a 0.5 (1 - 2.abs(cr))/6
• b 0.5 - (1 - 2.abs(cr))/6
• Aux a b r
• Else
• ?0
• endif

A value for ? function of the spatial variability
of P is compute in a way that the problems
describe in the earlier slide
rlt0gt ?0 in all schemes
rgt0
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1D channel test - TVD
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Code Trip

• The nuclear subroutine is ComputeAdvectionFace
• The advective flux in each face can depend of the
  property value in the follow compute points (i-2,
  i-1, I, i1)
• This subroutine compute for each face the
  coefficients a, b, c and d

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Test cases

• Already tested
• 1D and 2 D cyclic channel
• Symetric estuary
• Need more testing
• Fresh water cylinder
• Density front

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Fresh Water Cylinder - Exercise

• Depth 20 m, 20 layers
• 30 km x 30 km, dx 1 km
• Latitude 52º N
• Cylinder of 10 m deep and 3 km of radius. Outside
  the cylinder salinity is 34.85 psu Within the
  cylinder the salinity and the density variability
  is given by
• Run period 144 h
• Bottom rugosity 0
• Null turbulent diffusion or minimal values to
  avoid instabilities
• A relaxation boundary condition for the water
  level and salinity. Relax in the boundary the
  water level to 0 and salinity to 34.85 psu. A
  four-point-wide relaxation zone

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Conclusions

• The TVD with a Superbee limitation looks to be so
  far the best method. However, in extreme cases it
  generates wiggles

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Future Work

• Introduce the cross-derivates in the Quick and
  Quickest methods.

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Bibliography

• Pietrzak, J. (1997). The use of TVD limiters for
  forward-in-time upstream-biased advection schemes
  in ocean modeling. Monthly Weather Review. Volume
  126, 812-830, 1997 .
• Burchard, H., and K. Bolding, GETM - a general
  estuarine transport model. Scientific
  documentation, Tech. Rep. EUR 20253 EN, European
  Commission, 2002.
• James I.D. (1996). "Advection schemes for shelf
  sea models." Journal of Marine Systems 8
  237-254.
• James, I. D. (2000). "A high-performance explicit
  vertical advection scheme for ocean models how
  PPM can beat the CFL condition." Applied
  Mathematical Modelling, 24(1) 1-9.
• Tartinville, B., E. Deleersnijder, P. Lazure, R.
  Proctor, K.G. Ruddick and R.E. Uittenbogaard
  (1998). "A coastal ocean model intercomparison
  study for a three-dimensional idealised test
  case." Applied Mathematical Modelling, 22(3)
  165-182.
• Shchepetkin, A., J.C. McWilliams, 1998
  Quasi-monotone advection schemes based on
  explicit locally adaptive dissipation.  Monthly
  Weather Review, 126, 1541-1580.